The main aim of this proposal is to continue research in the following four areas (i) theoretical aspects of ascertainment sampling, in particular the development of ascertainment-assumption-free methods for pedigrees, for continuous characters and in regression problems, and to apply the theory to the estimation of parameters of genetic diseases; (ii) developments associated with the sampling theory of selectively neutral alleles, in particular the theory concerning age-order of alleles: (iii) practical aspects of the concept of a "minimum viable population size", applied to zoo and similar small populations, focussing on both genetic and demographic concepts; (iv) further development of the theory of random functions, in particular as it relates to population genetics. Since the proposed research is theoretical rather than experimental, there is no experimental design in the usual laboratory sense. The strategy of the research is to confirm theoretical predictions using simulated and real data using standard statistical methods. Computing of both mathematical functions and simulated "data" will be used throughout.